Computing Classical Escape Rates from Periodic Orbits in Chaotic Hydrogen

Placing a hydrogen atom in parallel electric and magnetic fields results in chaotic classical trajectories and complex quantum resonances. We are interested in studying the exponential escape of classical trajectories to infinity which model ionizing electron trajectories. This escape rate can be computed using periodic orbit theory, which converges absolutely with the inclusion of all periodic orbits. The theory of heteroclinic tangles is used to create a finite Markov partition of phase space and acquire a complete set of orbits up to a topological length of ten. This modest set of orbits is shown to be sufficient to accurately compute the escape rate for a range of energy values. The same set of orbits can be utilized in a semiclassical context to compute complex quantum resonances. In the fully quantum case, each energy eigenstate decays in time in a similar way to the classical ensemble of trajectories escaping in time. Relating classical trajectory ensembles to quantum eigenstates in this way allows for periodic orbits to be used to probe the classical-quantum transition.